Turbulent magnetic fields in the quiet Sun: implications of Hinode observations and small-scale dynamo simulations
Jonathan Pietarila Graham, Sanja Danilovic, Manfred Schuessler
aa r X i v : . [ a s t r o - ph ] M a r Turbulent magnetic fields in the quiet Sun: implications of
Hinode observations and small-scale dynamo simulations
Jonathan Pietarila Graham, Sanja Danilovic, and Manfred Sch¨ussler
Max-Planck-Institut f¨ur Sonnensystemforschung, 37191 Katlenburg-Lindau, Germany
ABSTRACT
Using turbulent MHD simulations (magnetic Reynolds numbers up to ≈ Hinode observations, we study effects of turbulence on measuring the solarmagnetic field outside active regions. Firstly, from synthetic Stokes V profiles forthe FeI lines at 6301 and 6302 ˚A, we show that a peaked probability distributionfunction (PDF) for observationally-derived field estimates is consistent with amonotonic PDF for actual vertical field strengths. Hence, the prevalence of weakfields is greater than would be naively inferred from observations. Secondly,we employ the fractal self-similar geometry of the turbulent solar magnetic fieldto derive two estimates (numerical and observational) of the true mean verticalunsigned flux density. We also find observational evidence that the scales ofmagnetic structuring in the photosphere extend at least down to an order ofmagnitude smaller than 200 km: the self-similar power-law scaling in the signedmeasure from a
Hinode magnetogram ranges (over two decades in length scalesand including the granulation scale) down to the ≈
200 km resolution limit. Fromthe self-similar scaling, we determine a lower bound for the true quiet-Sun meanvertical unsigned flux density of ∼
50 G. This is consistent with our numerically-based estimates that 80% or more of the vertical unsigned flux should be invisibleto Stokes − V observations at a resolution of 200 km owing to the cancellation ofsignal from opposite magnetic polarities. Our estimates significantly reduce theorder-of-magnitude discrepancy between Zeeman- and Hanle-based estimates. Subject headings:
Sun: magnetic fields — turbulence — MHD — techniques:polarimetric
1. Introduction
Determining the strength of the magnetization of the “quiet” Sun is tied to the ques-tion of how much flux resides at small scales. This is important, for example, in determining 2 –the energy budget available for chromospheric heating. Observationally, two methods areemployed to constrain the solar magnetic field: the Hanle and Zeeman effects. The Hanleeffect measures (in principle) the mean magnetic field strength, h| B |i , as there are no can-cellation effects, but quantitative interpretation requires assumptions about the probabilitydistribution function (PDF) of the turbulent magnetic field. The Hanle de-polarization ismeasured in stronger lines formed in the mid- to upper-photosphere and estimates are madeof h| B |i ∼
130 G, the field residing primarily in the intergranular lanes (Trujillo Bueno et al.2004). The Zeeman effect measures the longitudinal component (via Stokes V ) and trans-verse components (via Stokes Q and U ) of the magnetic field but suffers from cancellationeffects. Hence, it is “blind” to any “hidden” mixed-polarity flux at scales smaller than theresolution limit of an instrument. However, the benefit of Zeeman measurements is thattheir interpretation requires no assumption about the turbulent PDF and measurements canactually be used to determine the PDF on scales larger than the resolution limit. Someattempts to incorporate the effects of cancellation into Zeeman-based Stokes inversions uti-lize the micro-structured field hypothesis (Sanchez Almeida et al. 1996). However, typicalestimates based on the longitudinal Zeeman effect give h| B z |i ∼
10 G for the mean unsignedvertical flux density (see Table 3 in Bello Gonz´alez et al. 2009 for a review of the spread ofrecent Zeeman-based estimates). These values are significantly smaller than the Hanle-basedestimates. The discrepancy between the results from Hanle ( ∼
100 G) and Zeeman ( ∼
10 G)measurements is not unexpected since the Zeeman observations see only the resolved fluxwhile the Hanle interpretation depends on assumptions about an unknown PDF.In this work, we ask the question “How can the turbulent fractal geometry of the mag-netic field in the solar photosphere be accounted for in interpretations of Zeeman-basedobservations?” We attempt to derive the spectral/fractal properties from high resolution(0 ′′ .
3) Zeeman observations with the
Hinode spectro-polarimeter (SP) (Kosugi et al. 2007;Tsuneta et al. 2008; Lites et al. 2001) and high resolution (down to 4 km ≈ ′′ . Re M ≈ MURaM code (V¨ogler 2003; V¨ogler et al. 2005). First, we clarify the consistency of PDFs de-rived from observations and simulations, respectively. Zeeman observations typically showPDFs of quantities derived from Stokes V which can be described as a peaked function(Khomenko et al. 2005; Lites et al. 2008b). We demonstrate that the peaked PDFs fromStokes V measurements and the monotonic PDFs typically reported from numerical sim-ulations are, in fact, compatible. Failure to take into account this observational bias canlead to a gross underestimation of the prevalence of weak fields and, consequently, to incor-rect estimates of the mean magnetic field strength and magnetic energy density in the solarphotosphere.Second, we extrapolate the results obtained with observations and simulations (based 3 –on their respective fractal properties) to scales below the resolution limits to estimate theamount of “hidden” flux. To do this, we exploit the fact that the fractal geometry is in-timately connected to power-law scaling relations such as N ( l ) ∝ l − D f . Here, N ( l ) is thenumber of boxes of edge length l covering a fractal set (such as all pixels with apparentmagnetic flux above some threshold) and D f is the fractal dimension. It has long beenknown that the distribution of plage magnetic field is such a statistically self-similar fractal(Schrijver et al. 1992; Balke et al. 1993). As the threshold can influence the fractal dimensioninferred, however, the geometrical structure is more complicated than a simple fractal. In thiscase, the fractal concept must be generalized by adding a measure defined by the absolutevalue of the net magnetic flux through each box of edge length l . This measure also displaysself-similarity (a power-law scaling) for the solar quiet Sun network magnetic flux (down to0 ′′ . ′′ (see also S´anchez Almeida et al. 2003). For these mixed-polarity fields it isnecessary to further generalize the fractal concept to a signed measure. Here, the power-lawscaling exponent is the cancellation exponent (Ott et al. 1992). If this self-similar power-lawscaling extends below the observational resolution, small scale cancellation will occur andcorrect values of the mean field strength and energy cannot be established. As was pointedout by Lawrence et al. (1993), such a signed measure could also be employed to extrapolatemoments of the magnetic field below resolvable limits. Until now, no attempt has been madeto employ self-similar scaling to estimate the total cancellation, and, hence, the true meanquiet-Sun magnetic field strength.
2. Data and Methods
We use
MURaM simulations (V¨ogler 2003; V¨ogler et al. 2004, 2005) for a rectangulardomain of horizontal extent 4 . × .
86 Mm and a depth of 1 . FeI lines at 6301.5 ˚A and 6302.5 ˚A) ascalculated from a snapshot of a run with non-grey radiative transfer. Owing to the computa-tional expense of such a run, it was started from a snapshot from the statistically stationarystate of Run C (see Table 1) and then run for approximately one convective turnover time, 10minutes. Stokes V , Q , and U profiles were then computed in one dimension (1D) assuminglocal-thermodynamic-equilibrium (LTE) and using the STOPRO code in the
SPINOR package(Solanki 1987; Frutiger et al. 2000). We will concentrate in this work, however, on Stokes V observations (and synthetic observations) at disk center. This allows us to avoid line-of-sighteffects, to identify the longitudinal component of the magnetic field as its vertical component,and to avoid the difficult estimation of the cancellation properties of Stokes Q and U .Observational data is obtained from the spectro-polarimeter (SP, Lites et al. 2001) ofthe Solar Optical Telescope (SOT, Tsuneta et al. 2008) on the Hinode satellite (Kosugi et al.2007). One set, a spatial map, consists of 2048 scans taken on March 10, 2007 (11:37:36– 14:36:48 UT), in the “normal mode” (exposure time of 4 . . ′′ and pixel size along the slit of 0 . ′′ . The data set covers a quiet Sun region at thedisk center, over the large field of view of 324 ′′ × ′′ . The second set, a “deep mode” timeseries, consists of a 103 steps at disk center, each with an effective exposure time of 67 . SolarSoft procedure sp prep whichcalculates the wavelength-integrated Stokes V ( V tot), V tot = sgn( V b ) | R λ λ b V ( λ ) dλ | + | R λ r λ V ( λ ) dλ | I c R λ r λ b dλ , (1)where sgn( V b ) is the sign of the blue peak, λ is the line center, and λ r,b = λ ±
30 pm(Lites et al. 2008b). The procedure includes also a Milne-Eddington-based calibration of the V tot from both lines into a measure of longitudinal “apparent flux density”, B Lapp (Lites et al. 2008a). This calibration is tailored to retrieve the field value from weak andnoisy internetwork signals. It assumes that magnetic structures are spatially resolved (fillthe resolution element) and does not take into account the magnetic field variations over theheight range where lines are formed. Consequences of the latter assumption are studied inthe next section, when B Lapp , obtained from Stokes V profiles synthesized from simulations,is compared with the vertical component of the actual magnetic field.
3. Results3.1. PDFs
A marked difference exists between the PDFs inferred from Zeeman polarimetry (e.g.,Khomenko et al. 2005; Lites et al. 2008b) and the PDFs from numerical computations (e.g.,Socas-Navarro & S´anchez Almeida 2003; V¨ogler & Sch¨ussler 2007). For example, in Fig. 1we present the PDFs from the
Hinode “normal mode” map magnetogram (apparent verticalmagnetic flux density, dashed line) and of
MURaM simulation Run C-NG (average verticalmagnetic field, solid line). The polarimetric observation peaks at B Lapp ≈ B z ) monotonically decreaseswith increasing vertical field strength (instead of possessing a peak). Our approach will beto assume the distribution of field strengths from turbulent MURaM simulations and examinethe consequences of such a distribution on Stokes V observations.Though noise, resolution, and other instrumental factors are important in any real obser-vation, we first address the question assuming a “perfect” instrument. Using the syntheticprofiles from Run C-NG, we calculate V tot with Eq. (1) and determine B Lapp followingLites et al. (2008b). In Fig. 2, the derived B Lapp signal versus B ave, the vertical magneticfield strength averaged over the height range corresponding to log τ ∈ [ − . , . B Lapp of r = 0 .
92 and its co- 6 –efficient of linearity, B Lapp ≈ . B ave, which is consistent with the calibration of Lites et al.(2008b). This height range also encompasses most of the formation height of the FeI linesat 6301 and 6302 ˚A. Though B Lapp and B ave are well correlated, there is a large scatter.We note, also, that changing the range to log τ ∈ [ − , .
1] does not significantly affect thecorrelation, r . This indicates that most of the Stokes V signal is generated in deeper layers(Orozco Su´arez et al. 2007).In Fig. 1, we present a comparison between the PDFs of B Lapp as derived from thesynthetic Stokes V profiles (dash-dotted line) and B ave (solid line). PDF( B ave) monotoni-cally decreases with increasing field strength while PDF( B Lapp) shows a peak near 1 G and astrong decline towards smaller field strengths. The PDFs for maximum Stokes V amplitudeand total circular polarization are qualitatively similar to that shown for B Lapp. PDFs forthe vertical magnetic field from different volumes and 2D cross sections from all simulationruns, chosen either by height or by optical depth, show similar PDFs to that shown for B ave.That is, the monotonically decreasing distribution is a robust feature of the vertical magneticfield when sampled by geometrical height, optical depth, or by averaging over the verticaldirection. The difference between observations and simulations is caused by the radiativetransfer that produces circular polarization from longitudinal magnetic field.The above result shows that caution is needed when interpreting the distribution ofStokes V signal in order to avoid a drastic underestimation of the occurrence of weak field.This caution naturally extends to moments of the distribution such as mean vertical fluxdensity or mean vertical magnetic energy density. For example, B Lapp and B ave are verywell correlated with a coefficient of linearity of unity, but their averages, D | B Lapp | E = 6 . h| B ave |i = 5 . V , even in the absence of noise (note that this is close to the 20% loss found inS´anchez Almeida et al. 2003).To understand in more detail how radiative transfer affects contribute to a peaked PDF,we examine a few selected V − profiles. Pixels with weak B ave must be generating strong B Lapp signals for PDF( B Lapp) to become peaked. Indeed, Fig. 3 shows this is the case. Thereare many pixels for which | B ave | < . | B Lapp | > | B Lapp | < . | B ave | is always less than 4 G. In Fig. 4, we examine one case ofhow weak B ave can be associated with strong B Lapp. For τ ∈ [0 . , B ave is nearly zero. However, because of the velocity gradient, the 7 –contributions to the Stokes − V profile from the positive and negative magnetic polaritiesare Doppler-shifted with respect to each other. For this reason, the Stokes V signal is notcancelled. This is further illustrated in Fig. 5 where we plot the mean B Lapp for all pixelswith | B ave | < . B Lapp. In effect, | V tot | is some combination of thevertical mean of B z and the vertical mean of | B z | (depending on v z ( z )) and, consequently,the PDF of B Lapp does not correspond to PDF( B z ) from any geometrical height, opticaldepth, or volume. Such a failing of B Lapp to accurately represent B z cannot be capturedusing the Milne-Eddington approximation (used to calibrate B Lapp), which has no gradientsby definition.To examine the effect of noise on the PDF, we consider synthetic Stokes V profileswith noise added at a polarization precision of 1 . × − (similar to that of the Hinode observations) in determining B Lapp. The PDF of this noisy synthetic observation is shownas a dotted line in Fig. 1 and closely resembles the observational PDF for signals weakerthan a few Gauss. Note that, the noise accentuates the peak in the PDF even further. Inexamining Eq. (1) for V tot ( B Lapp is a nearly-linear function of V tot), we see that by takingthe absolute value of the blue and red lobes separately the effect of noise becomes the sumof two non-negative measurement errors. That is, V measuredtot = V truetot + | ǫ b | + | ǫ r | , (2)where ǫ b and ǫ r are the measurement noise in the blue and red lobes (e.g., ǫ b ≡ P N b i =1 ǫ i /N where ǫ i are the random variables associated with the measurement noise in each wavelengthbin). Assuming these two random variables, ǫ b and ǫ r have Gaussian distributions, theirseparate PDFs for their absolute values will peak at zero. However, the PDF of the sum oftheir absolute values will peak at a non-zero value due to reduced likelihood that | ǫ b | and | ǫ r | are small simultaneously: the PDF of the sum of two independent random variables isthe convolution of their individual PDFs, P ( ǫ ) = 2 πσ b σ r Z ǫ e − ( ǫ − ξ ) / σ b e − ξ / σ r dξ (3)for ǫ b , ǫ r Gaussian and ǫ = | ǫ b | + | ǫ r | . Because of taking the absolute values, the individualPDFs are zero for negative values (this sets the limits of integration for Eq. (3)). Hence,their convolution is zero at zero and peaks instead for some finite positive value. Assuming ǫ b and ǫ r have identical identical standard deviation σ = √ · . B Lapp, Eq. (3), is given by the solution 8 –to Bσ Z B/ e − ξ dξ − e − B / = 0 . (4)This predicts a peak in the PDF at B Lapp ≈ B z ) is qualitatively consistent with observations.In Fig. 6, we use the 67 . σ = √ · . B Lapp ≈ ≈ . MURaM B Lapp with equivalent noise level (dotted line) matches thelocation of the peak and the PDF to the left of the peak. In fact, if we generate B Lapp frompure white noise for Stokes V (standard deviation of 3 × − ), we find its PDF (plus signs)predicts well both the location of the peak and the shape of the weak-signal portion of thePDFs. This strongly suggests that the observational peak is dominated by noise.We also find (see Fig. 6) that cancellation of opposite polarity fields in a resolutionelement alters the PDF and renders it useless for computations of the mean unsigned fluxdensity and other moments. This is evidenced by a comparison of PDFs from the noisysynthetic MURaM B Lapp without (dotted line) and with spatial degradation by a theoreticalpoint spread function (PSF, see Danilovic et al. 2008 for details) for
Hinode ’s optical systemand rebinned to
Hinode pixel size (diamonds). Because of the importance of the cancellationon the PDFs, PDFs may not be used to infer the true mean unsigned vertical flux density.
Turbulence gives rise to a statistically self-similar fractal pattern of the magnetic field(within the inertial range) – the field retains the same degree of complexity of distinctstructures regardless of the scale at which it is observed (see, e.g., Constantin & Procaccia1992; Brandenburg et al. 1992). In this section, we show that this also applies to solar surfacemagnetic fields and we use this self-similarity to estimate the portion of unsigned vertical fluxunobservable at a given resolution. To begin with, we examine the cancellation propertiesof the magnetic field itself using a series of high-resolution
MURaM dynamo simulations. Thisillustrates how the turbulent nature of the magnetic field limits measurement under the soleconsideration of spatial resolution and in the absence of other observational constraints. Asit separates the statistics of the field itself from observational constraints, the study alsoallows us to extrapolate the results to realistic solar magnetic Reynolds numbers. 9 –Extending the ideas of singularity in probability measures for self-similar fractal fieldsto signed fields, Ott et al. (1992) introduced the cancellation exponent for studying the self-similar sign oscillations on very small scales in turbulent flows. For our application, theirpartition function, χ ( l ), measures the portion of the flux remaining after averaging overboxes of edge length l , χ ( l ) ≡ P i (cid:12)(cid:12)(cid:12) R A i ( l ) B z da (cid:12)(cid:12)(cid:12)R A | B z | da (5)where {A i ( l ) } ⊂ A is a hierarchy of disjoint subsets of size l covering the entire domain, A . In our case, we call the function χ the cancellation function since it measures the fluxcancellation at a given length-scale l . If the magnetic field is self-similar (for scales muchlarger than the dissipation scale), we expect a power-law χ ( l ) ∝ l − κ , (6)where κ is called the cancellation exponent . It is related to the characteristic fractal dimensionof the magnetic field structures on all scales, D f , by κ = ( d − D f ) / d = 2 is the embedding Euclidean dimension of the solar surface (Sorriso-Valvo et al.2002). An improved method to determine χ ( l ) using a Monte Carlo box counting techniquewas proposed by Cadavid et al. (1994). Its advantages include better counting statistics when l is a large fraction of the edge length of the domain A , applicability to non-square pixels,and less sensitivity to the accidental placement of larger flux patches (e.g., network elements)with respect to the partitioning. For our simulation data, the Monte Carlo technique provedas accurate as rigid partition boxes but led to a significant reduction of the noise in χ ( l ):it averages over many partitionings and allows a more faithful representation of the fielddistribution (Cadavid et al. 1994). We use this technique for the results shown below.The height range that corresponds (in a horizontally averaged sense) to log τ ∈ [ − , . § τ ∈ [ − . , −
2] to the Stokes V signal is in-significant) is z ∈ [210 , z = 0 corresponds to the continuum optical depth τ = 1at 500 nm). For this height range we compute the averaged cancellation functions, χ ( l ), for MURaM dynamo simulations with magnetic Reynolds numbers ranging from Re M ≈ Assuming the field is smooth (correlated) in D f dimensions and uncorrelated in the other d − D f dimensions, the smooth dimensions contribute to the sum of vertical fluxes proportional to their area whilethe integral of an uncorrelated field contributes proportional to the square root of its area (random process).Eq. (7) then follows (Sorriso-Valvo et al. 2002).
10 – Re M ≈ χ ( l ) = 1 at the resolution of the simulation since thereare no smaller scales for the computation. Furthermore, we expect dissipation to stronglyaffect χ ( l ) for the smallest decade of scales (analytically, its slope must go to zero). Also, asour dynamo simulations have zero signed total flux, χ (4 .
86 Mm) = 0 and we would expectscales down to approximately 490 km to be affected by this constraint. Only for smallerscales should we be able to observe a turbulent scaling. However, very little room is leftbetween these two constraints so that no clear power-law scaling is observed for any of thesimulations (see Fig. 7 for one example).Since the dissipation scale of magnetic energy, l η , decreases with increasing Re M , forfixed l , χ ( l ) decreases with increasing magnetic Reynolds number (fluctuations at smallerscales increase the total cancellation). This is emphasized in Fig. 8, where we plot the valueof the cancellation function for l = 200 km (corresponding roughly to Hinode
SP’s angularresolution of 0 ′′ .
3) versus Re M . We can fit a power law and extrapolate to the results wewould expect from a MURaM simulation at solar Re M (which must be estimated). FromKovitya & Cram (1983), we estimate the magnetic diffusivity for log τ = 0, η ∼ cm s − .The driving of the small-scale dynamo is mainly subsurface where η is roughly 100 timessmaller ( η ∼ cm s − , cf. Spruit 1974). For an upper limit of χ (200 km), we employ themore conservative estimate: η ∼ cm s − . Taking the forcing scale to be the granulationscale, L ∼ v rms ∼ − from the simulation, we find Re M ≡ L v rms η ∼ · . (8)For this magnetic Reynolds number, our extrapolation yields χ (200 km) ∼ .
2. This indi-cates that with a perfect observation at this spatial resolution and assuming that the
MURaM simulation faithfully reproduces the solar conditions, we should multiply an observation bya factor of 5 to obtain the true mean vertical unsigned flux density of the quiet-Sun inter-network. It is also suggested by Fig. 8 that χ (200 km) decreases with decreasing magneticPrandtl number, P M ≡ ν/η where ν is the kinematic viscosity and η the magnetic diffusivity.As the magnetic Prandtl number of the Sun is much less than that of the simulations, weexpect that χ (200 km) . . B z and for B Lapp inferred from Run C-NG are shownin Fig. 7. We see that the two functions are essentially equivalent. This demonstrates anexcellent correspondence between the cancellation of the field itself and the signal derivablefrom observations (excluding instrumental effects). Therefore, we may take the cancellationof B Lapp as a proxy for the cancellation of B Z . This we now do.We present the normalized cancellation function, χ ( l ) /χ (1 Mm), for the Hinode
SPobservation in Fig. 9. Without knowing the value of the true unsigned vertical flux, the 11 –denominator in Eq. (5), the value of the cancellation function can only be normalized tosome arbitrary scale. We find a self-similar power-law over two decades in length scales,demonstrating the multifractal geometry of the turbulent quiet-Sun magnetic field. This issomewhat surprising as the dominant granulation pattern at scales near 1 Mm might havebeen expected to affect the cancellation scaling. The cancellation exponent of the scalingis κ = 0 . ± .
01. This exponent predicts a 20% increase in the observed mean unsignedvertical flux density with a doubling of resolution in agreement with the difference in fluxdensities found between ground and space-based telescopes (Lites et al. 2008b). Note alsothat the power-law behavior holds down to the two-pixel scale. This is a clear indication ofcancellation extending to smaller scales than resolved by
Hinode (Carbone & Bruno 1997;Sorriso-Valvo et al. 2004). Compare this, for example, to the simulation case in Fig. 7 (alsosee Fig. 3(b) of S´anchez Almeida et al. 2003) where dissipation is seen to affect a strongturnover in χ ( l ) for the smallest decade of scales. As the observation is not so affected, wemay safely conclude that the smallest scale of magnetic structuring is at least one decadesmaller than the Hinode
SP resolution limit. The scales of magnetic structuring in thephotosphere must therefore extend to at least an order of magnitude smaller than 200 km.From Eq. (7), we see that our result corresponds to D f = 1 . ± .
02 for the fractaldimension of the quiet Sun internetwork magnetogram. Within uncertainties, this is thesame dimension as for solar plage regions, D f = 1 . ± .
05 (Balke et al. 1993). This mightindicate that some similar mechanisms are at play in solar plage and quiet Sun internetwork.For the cancellation exponent of network magnetic fields, values of κ ∼ . κ ∼ .
12 (Cadavid et al. 1994) have been reported, but without an estimate ofthe uncertainties.Recent work has highlighted the sensitivity of fractal dimension (perimeter-area) esti-mators to pixelization and resolution (Criscuoli et al. 2007). By using a signed measure,however, we avoid difficulties inherent to fractal dimension estimations using bi-level imagesin general and the perimeter-area method, specifically. Nonetheless, we have tested the sen-sitivity of the cancellation exponent to reducing our resolution by theoretical point spreadfunctions for apertures 1/2 and 1/4 that of the
Hinode
SOT (50 cm). We find the slope of χ ( l ) to be robust in these cases for lengths exceeding 30 pixels. There is no change in thepower law for almost one decade of length scales (3-20Mm). We therefore conclude that ourestimation, κ = 0 . ± .
01 is robust and insensitive to pixelization and resolution effects.As pointed out by Lawrence et al. (1996), however, because of what they call “resolution-limited asymptotics”, different definitions of fractal dimension can give different values atfinite resolution. For this reason, our value D f = 1 . ± .
02 might differ from a well-resolvedperimeter-area estimate. 12 –Using the self-similar power law derived from Fig. 9, we may estimate the true meanunsigned vertical component of the magnetic field (hereafter, “mean unsigned vertical fluxdensity”) in the quiet-Sun photosphere, h| B z |i . Below the magnetic dissipation scale, l η ,there is no cancellation: χ ( l η ) ≡
1. This, together with the self-similarity relation, Eq. (6),gives h| B z |i = h| B z |i l η = h| B z |i l · (cid:18) ll η (cid:19) κ , (9)where l is any scale in the inertial range, h| B z |i l is the mean absolute value of the verticalcomponent of the field measured at that resolution ( l ), h| B z |i l ≡ P i (cid:12)(cid:12)(cid:12) R A i ( l ) B z da (cid:12)(cid:12)(cid:12)R A da = χ ( l ) · h| B z |i , (10)and h| B z |i is given by h| B z |i ≡ R A (cid:12)(cid:12)(cid:12) B z (cid:12)(cid:12)(cid:12) da R A da . (11)Lites et al. (2008b) report h| B z |i . ≈ . l ≈ .
11 Mm (approximate
Hinode
SPpixel size) is below the SOT resolution limit, however, we rebin B Lapp to l ≈ .
22 Mm pixelsto find h| B z |i . ≈ . h| B z |i ≈ . · (cid:18) .
22 Mm l η (cid:19) . . (12)Estimating the magnetic dissipation scale is not straight-forward. As we have shownthat observationally it is unresolved, we are left to rely on a phenomenological estimate.Kolmogorov phenomenology predicts (see, e.g., Frisch 1995) l η ≈ L Re − / M where L is a largecharacteristic scale, such as the granulation scale. Using Re M ∼ · , derived previously,we estimate l η ∼
80 m. For the dissipative range, power-law scaling for χ ( l ) will not applyand the slope of the cancellation function will approach zero. To provide a lower boundto the solar mean unsigned vertical field, we should then be conservative by ignoring anycancellation in the first decade of scales. Hence, we use l η = 800 m in Eq. (12) to estimatethe true mean unsigned vertical flux density to be h| B z |i &
46 G. This means that, at aresolution of 200 km, at most one quarter of the unsigned vertical flux is observable.
4. Discussion
Our estimates suggest that three-quarters or more of the vertical unsigned magneticflux is cancelled at the resolution of
Hinode . Hanle-based estimates suggest h| B |i ∼
130 G 13 –(Trujillo Bueno et al. 2004) while Zeeman-based estimates suggest h| B z |i ∼
10 G (see Table3 in Bello Gonz´alez et al. 2009). Note that even with estimation of the cancellation, there re-mains almost a factor of 3 difference between reported Hanle estimates and the Zeeman-basedestimates we present. However, we have considered only one component of a vector quantitywhile the Hanle-based estimates are sensitive to the magnitude of that vector. Recent ob-servations (Lites et al. 2008b) and simulations (Steiner et al. 2008; Sch¨ussler & V¨ogler 2008)suggest that horizontal fields are on average a factor of 5 stronger than vertical fields. There-fore, our estimate of h| B z |i &
46 G coupled with an even stronger mean horizontal field isconsistent with the Hanle-based estimate. Another observational discrepancy lies in deter-mining the mean “location” of the fields. Trujillo Bueno et al. (2004) interpret scatteringpolarization from molecular C to indicate the mean field strength is weak ( ∼
10 G) over thebright granules, so that the turbulent field inferred from the Hanle measurements should beconcentrated in the intergranular lanes. This should be compared to the location of stronghorizontal fields not in the lanes but near the edges of granules (Lites et al. 2008b). In re-solving the details over the location and strength of the mean components of the magneticfield, future work should also address the cancellation statistics of the horizontal field (andthe linearly-polarized Stokes signals Q and U ) as well as the effect on Stokes V presentedhere.
5. Conclusion
On the basis of surface dynamo simulations, we have demonstrated that the PDF gen-erated from the Stokes V spectra are not necessarily equivalent in form to that of the PDFof the vertical component of the underlying magnetic field. The PDF for Stokes V showsa reduction of likelihood for weak vertical magnetic field compared to the PDF of the fielditself. This effect is not due to a reduction in horizontal resolution, but is caused by a com-bination of vertical radiative transfer through a turbulent fluid (via the Doppler effect) andnoise. That is, any systematic sampling (by geometrical height, optical depth, or volume) of B z from the simulation yields a monotonic PDF, but due to Doppler shifts between differentatmospheric heights, the Stokes V signal is not such a systematic sampling. Consequently,the PDF of Stokes V field estimates do not accurately represent the PDF of the actual verti-cal magnetic field even in the absence of noise. Additionally, for two different levels of noise(“normal mode” and “deep mode”) we have demonstrated that the peak in the observational Rather than assuming that a turbulent magnetic field possesses a delta-function PDF which leads to the ∼
60 G estimate in Trujillo Bueno et al. (2004), we take here the ∼
130 G estimate from their assumption ofan exponential PDF.
14 –PDF is dominated by the influence of noise. Because of these two effects, a monotonic PDFfor the field can result in a peaked PDF in observations and the assumption that PDF( B z )can be uniquely derived from Stokes V observations becomes dubious.From the cancellation function for a Hinode observation of the apparent longitudinalflux density, we have demonstrated that the multi-fractal self-similar pattern of the quiet-Sun photospheric magnetic field covers two decades of length scales down to the resolutionlimit, 200 km. This constitutes observational evidence that the the smallest scale of magneticstructuring in the photosphere is at least an order of magnitude smaller than 200 km. Thepower law also allows us to constrain the quiet-Sun true mean unsigned vertical flux density.We estimate the lower bound to be ≈
46 G. Estimates based solely on our numerical simula-tions suggest that the vertical unsigned flux at
Hinode ’s resolution should be multiplied by 5to obtain the true vertical unsigned flux (i.e., ∼
50 G). These two results are consistent andsuggest that the order of magnitude disparity between Hanle and Zeeman-based estimatesmay be fully resolved by a proper consideration of the cancellation properties of the fullvector field.
Acknowledgments
The authors would like to acknowledge fruitful discussions with S. Solanki, A. Pietarila,and R. Cameron.
Hinode is a Japanese mission developed and launched by ISAS/JAXA,collaborating with NAOJ as a domestic partner, NASA and STFC (UK) as internationalpartners. Scientific operation of the Hinode mission is conducted by the Hinode scienceteam organized at ISAS/JAXA. This team mainly consists of scientists from institutes inthe partner countries. Support for the post-launch operation is provided by JAXA andNAOJ (Japan), STFC (U.K.), NASA, ESA, and NSC (Norway).
Note added in proof
We would like to point out the correlations between our conclusions and the worksof Sanchez Almeida et al. (1996) and S´anchez Almeida & Lites (2000) who postulate struc-turing of the magnetic and velocity fields on scales much smaller than 100 km. They findthat synthetic profiles generated by 3-component Milne-Eddington atmospheres re-producethe observed Stokes- V asymmetries found in 1 ′′ resolution observations. Though they didnot estimate the undetected photospheric magnetic flux, the results indicated a significantfraction remaining undetected. Dom´ınguez Cerde˜na et al. (2006) assumed the quiet-SunPDF can be approximated by a linear combination of the PDF inferred from Zeeman ob-servations and a log-normal distribution (accounting for the observed Hanle depolarization).They determined that the Hanle and Zeeman signals are consistent with a single PDF with h| B |i &
100 G (see also S´anchez Almeida et al. 2003). S´anchez Almeida (2006) assuming 15 –that numerical simulations of magnetoconvection with no dynamo action (20 km horizon-tal resolution) had achieved the asymptotic rate of magnetic energy dissipation, derivedan estimate for the unsigned magnetic flux contained in unresolved scales; in our nota-tion their finding is χ (100 km) ∼ . ∼ .
36. Finally,S´anchez Almeida (2008) using observational data from various sources plot h| B z |i l versus l (their Fig. 1). The data are compared to a line, the slope of which corresponds to κ = 1,i.e., the result for white noise (Vainshtein et al. 1994, also set D f = 0 in Eq. (7)). There is,however, a large scatter about this line suggestive of either a large uncertainty in κ or anelement of randomness in the calibration issues between the various data used. REFERENCES
Balke, A. C., Schrijver, C. J., Zwaan, C., & Tarbell, T. D. 1993, Sol. Phys., 143, 215Bello Gonz´alez, N., Yelles Chaouche, L., Okunev, O., & Kneer, F. 2009, A&A, 494, 1091Brandenburg, A., Procaccia, I., Segel, D., & Vincent, A. 1992, Phys. Rev. A, 46, 4819Cadavid, A. C., Lawrence, J. K., Ruzmaikin, A. A., & Kayleng-Knight, A. 1994, ApJ, 429,391Carbone, V. & Bruno, R. 1997, ApJ, 488, 482Constantin, P. & Procaccia, I. 1992, Phys. Rev. A, 46, 4736Criscuoli, S., Rast, M. P., Ermolli, I., & Centrone, M. 2007, A&A, 461, 331Danilovic, S., Gandorfer, A., Lagg, A., Sch¨ussler, M., Solanki, S. K., V¨ogler, A., Katsukawa,Y., & Tsuneta, S. 2008, A&A, 484, L17Dom´ınguez Cerde˜na, I., S´anchez Almeida, J., & Kneer, F. 2006, ApJ, 636, 496Frisch, U. 1995, Turbulence, The Legacy of A. N. Kolmogorov (Cambridge, UK: CambridgeUniversity Press)Frutiger, C., Solanki, S. K., Fligge, M., & Bruls, J. H. M. J. 2000, A&A, 358, 1109Khomenko, E. V., Mart´ınez Gonz´alez, M. J., Collados, M., V¨ogler, A., Solanki, S. K., RuizCobo, B., & Beck, C. 2005, A&A, 436, L27 16 –Kosugi, T., Matsuzaki, K., Sakao, T., Shimizu, T., Sone, Y., Tachikawa, S., Hashimoto,T., Minesugi, K., Ohnishi, A., Yamada, T., Tsuneta, S., Hara, H., Ichimoto, K.,Suematsu, Y., Shimojo, M., Watanabe, T., Shimada, S., Davis, J. M., Hill, L. D.,Owens, J. K., Title, A. M., Culhane, J. L., Harra, L. K., Doschek, G. A., & Golub,L. 2007, Sol. Phys., 243, 3Kovitya, P. & Cram, L. 1983, Sol. Phys., 84, 45Krivova, N. A. & Solanki, S. K. 2004, A&A, 417, 1125Lawrence, J. K., Cadavid, A. C., & Ruzmaikin, A. A. 1996, ApJ, 465, 425Lawrence, J. K., Ruzmaikin, A. A., & Cadavid, A. C. 1993, ApJ, 417, 805Lites, B. W., Elmore, D. F., & Streander, K. V. 2001, in Astronomical Society of the PacificConference Series, Vol. 236, Advanced Solar Polarimetry – Theory, Observation, andInstrumentation, ed. M. Sigwarth, 33–+Lites, B. W., Ichimoto, K., Kubo, M., & et al. 2008a, Sol. Phys., submittedLites, B. W., Kubo, M., Socas-Navarro, H., Berger, T., Frank, Z., Shine, R., Tarbell, T.,Title, A., Ichimoto, K., Katsukawa, Y., Tsuneta, S., Suematsu, Y., Shimizu, T., &Nagata, S. 2008b, ApJ, 672, 1237Orozco Su´arez, D., Bellot Rubio, L. R., & del Toro Iniesta, J. C. 2007, ApJ, 662, L31Ott, E., Du, Y., Sreenivasan, K. R., Juneja, A., & Suri, A. K. 1992, Physical Review Letters,69, 2654S´anchez Almeida, J. 2006, A&A, 450, 1199—. 2008, Ap&SS, 156S´anchez Almeida, J., Emonet, T., & Cattaneo, F. 2003, ApJ, 585, 536Sanchez Almeida, J., Landi degl’Innocenti, E., Martinez Pillet, V., & Lites, B. W. 1996,ApJ, 466, 537S´anchez Almeida, J. & Lites, B. W. 2000, ApJ, 532, 1215Schrijver, C. J., Zwaan, C., Balke, A. C., Tarbell, T. D., & Lawrence, J. K. 1992, A&A, 253,L1Sch¨ussler, M. & V¨ogler, A. 2008, A&A, 481, L5 17 –Socas-Navarro, H. & S´anchez Almeida, J. 2003, ApJ, 593, 581Solanki, S. K. 1987, PhD thesis, ETH Z¨urichSorriso-Valvo, L., Carbone, V., Noullez, A., Politano, H., Pouquet, A., & Veltri, P. 2002,Physics of Plasmas, 9, 89Sorriso-Valvo, L., Carbone, V., Veltri, P., Abramenko, V. I., Noullez, A., Politano, H.,Pouquet, A., & Yurchyshyn, V. 2004, Planet. Space Sci., 52, 937Spruit, H. C. 1974, Sol. Phys., 34, 277Steiner, O., Rezaei, R., Schaffenberger, W., & Wedemeyer-B¨ohm, S. 2008, ApJ, 680, L85Trujillo Bueno, J., Shchukina, N., & Asensio Ramos, A. 2004, Nature, 430, 326Tsuneta, S., Ichimoto, K., Katsukawa, Y., Nagata, S., Otsubo, M., Shimizu, T., Suematsu,Y., Nakagiri, M., Noguchi, M., Tarbell, T., Title, A., Shine, R., Rosenberg, W., Hoff-mann, C., Jurcevich, B., Kushner, G., Levay, M., Lites, B., Elmore, D., Matsushita,T., Kawaguchi, N., Saito, H., Mikami, I., Hill, L. D., & Owens, J. K. 2008, Sol. Phys.,249, 167Vainshtein, S. I., Sreenivasan, K. R., Pierrehumbert, R. T., Kashyap, V., & Juneja, A. 1994,Phys. Rev. E, 50, 1823V¨ogler, A. 2003, PhD thesis, G¨ottingen UniversityV¨ogler, A., Bruls, J. H. M. J., & Sch¨ussler, M. 2004, A&A, 421, 741V¨ogler, A. & Sch¨ussler, M. 2007, A&A, 465, L43V¨ogler, A., Shelyag, S., Sch¨ussler, M., Cattaneo, F., Emonet, T., & Linde, T. 2005, A&A,429, 335
This preprint was prepared with the AAS L A TEX macros v5.2.
18 –Fig. 1.— Probability distribution functions (PDFs) for magnetic field strengths and derivedfield proxies:
Hinode
SP “normal mode” map B Lapp (dashed line),
MURaM simulation B ave(see text, solid line), MURaM synthetic B Lapp (B derived from Stokes V , dot-dashed), and B Lapp including a noise level of 1 . × − (dotted). The PDFs of the synthetic observationsappear peaked although we have a monotonic distribution of vertical field strengths. 19 –Fig. 2.— B Lapp derived from
MURaM
Run C-NG versus B ave, the actual vertical magneticfield strength averaged over log τ ∈ [ − . , . r = 0 .
92. Note the large scatter. 20 –Fig. 3.— (Left) B Lapp versus B ave for B ave < . (Right) B ave versus B Lapp for B Lapp < . B Lapp computed from noiseless V-profiles). These plots indicate the bias thatstrong Stokes V signal can be associated with a pixel with weak averaged magnetic field,but seldomly vice-versa .Fig. 4.— (Left) B z (solid line) and v z (dashed line) versus optical depth, τ nm, and (Right) Stokes V profile for the pixel indicated by a diamond in Fig. 3 ( B Lapp = − . B ave = − . · − G). At log τ = 0 the positive and negative contributions to B ave havenearly cancelled (integrating downward). The Stokes V signal is stronger than would resultfrom a uniform 1 . · − G field but is asymmetric. Strong gradients lead to asymmetricprofiles but also to | B Lapp | ≫ | B ave | . 21 –Fig. 5.— Average B Lapp versus standard deviation of the fluctuations of the vertical velocityalong the (vertical) line-of-sight, σ v , for all pixels with | B ave | < . σ v before averaging. With strong velocity differences between different heights in the atmo-sphere, the total Stokes V signal increases as the Doppler-shifted absorption from positivelyand negatively oriented fields show less cancellation. 22 –Fig. 6.— PDFs for derived field proxies: Hinode
SP “deep mode” time series B Lapp (dashedline) and
MURaM synthetic B Lapp including a noise level of 3 × − (dotted). The effectsof cancellation due to finite spatial resolution are seen in the PDF of the synthetic signalincluding noise as well as spatial smearing from a theoretical PSF and rebinning to Hinode resolution (diamonds). As this represents a real loss of data, the true mean unsigned verticalflux density cannot be calculated from the observational PDF. Also shown is the result foremploying pure white noise with a standard deviation of 3 × − for Stokes V (+). 23 –Fig. 7.— Cancellation function, χ ( l ), versus scale, l for Run C-NG: B z (solid line) and B Lapp (simulated observation, dashed). The two are essentially equivalent, suggesting thatthe cancellation of B Lapp may be taken as a proxy for the cancellation of B z . 24 –Fig. 8.— Portion of flux remaining at l = 200 km, χ (200 km), versus magnetic Reynoldsnumber, Re M . Symbols are Run E (plus), Run C (asterisk), Run G (diamond), Run H(X), Run C-NG (triangle), and Run G-P (square)–see Table 1. For fixed l , χ ( l ) decreaseswith Re M and shows an approximate power-law relation with Re M as indicated by the fitteddashed line. Run C-NG and Run G-P are not included in the fit, but the effect of decreasedmagnetic Prandtl number leads to reduced χ (200 km). Taking this into consideration, alongwith extrapolation to solar values, Re M ∼ · , we estimate χ (200 km ) . .
2. 25 –Fig. 9.— Normalized cancellation function, χ ( l ) /χ (1 Mm), versus scale, l , from Hinode B Lappobservation. A self-similar power-law is abundantly clear for 2 decades of length scales downto the resolution limit of the observation (the fitted line is k = 0 . ± . MURaM simulation runs: shown are grid points, horizontal resolution,and magnetic Reynolds number, Re M . All runs except Run C-NG utilize grey radiativetransfer. In Run C-NG, opacity binning with 4 bins (V¨ogler et al. 2004) has been used toprovide non-grey radiative transfer. For all simulations no physical viscosity is imposed.Rather, numerical dissipative effects lead to an effective kinetic Reynolds number, Re (V¨ogler et al. 2005). To obtain a lower value of P M = Re M /Re , Run G-P uses themagnetic diffusivity used in Run C but at a higher resolution, hence higher Re .Simulation Computational Grid Horizontal Resolution Re M Run E 540 × ×
140 9 km ≈ × ×
140 7 . ≈ × ×
140 7 . ≈ × ×
200 5 km ≈ × ×
200 5 km ≈ × ×
350 4 km ≈≈